MinT - Best Known (222−15, 222, s)-Nets in Base 4

Best Known (222−15, 222, s)-Nets in Base 4

(222−15, 222, 4880885)-Net over F₄ — Constructive and digital

Digital (207, 222, 4880885)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (43, 50, 87401)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 17)-net over F₄, using
      - net defined by OOA [i] based on linear OOA(4⁴, 17, F₄, 3, 3) (dual of [(17, 3), 47, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(4⁴, 17, F₄, 2, 3) (dual of [(17, 2), 30, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁴, 17, F₄, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
        ovoid in PG(3,Â 4) [i]
    2. digital (39, 46, 87384)-net over F₄, using
      - net defined by OOA [i] based on linear OOA(4⁴⁶, 87384, F₄, 7, 7) (dual of [(87384, 7), 611642, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(4⁴⁶, 262153, F₄, 7) (dual of [262153, 262107, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(4⁴⁶, 262144, F₄, 7) (dual of [262144, 262098, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(4³⁷, 262144, F₄, 6) (dual of [262144, 262107, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(4⁰, 9, F₄, 0) (dual of [9, 9, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (157, 172, 4793484)-net over F₄, using
  - trace code for nets [i] based on digital (28, 43, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴³, 1198371, F₂₅₆, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(256⁴³, 8388598, F₂₅₆, 15) (dual of [8388598, 8388555, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴³, large, F₂₅₆, 15) (dual of [large, large−43, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(222−15, 222, large)-Net over F₄ — Digital

Digital (207, 222, large)-net over F₄, using

t-expansion [i] based on digital (204, 222, large)-net over F₄, using
- 5 times m-reduction [i] based on digital (204, 227, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²²⁷, large, F₄, 23) (dual of [large, large−227, 24]-code), using
    - 22 times code embedding in larger space [i] based on linear OA(4²⁰⁵, large, F₄, 23) (dual of [large, large−205, 24]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(222−15, 222, large)-Net in Base 4 — Upper bound on s

There is no (207, 222, large)-net in base 4, because

13 times m-reduction [i] would yield (207, 209, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (222−15, 222, s)-Nets in Base 4

(222−15, 222, 4880885)-Net over F4 — Constructive and digital

(222−15, 222, large)-Net over F4 — Digital

(222−15, 222, large)-Net in Base 4 — Upper bound on s

(222−15, 222, 4880885)-Net over F₄ — Constructive and digital

(222−15, 222, large)-Net over F₄ — Digital